Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 18.6s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[-0.5 \cdot \frac{c}{b}\]

C

double f(double a, double b, double c) {
        double r5860723 = b;
        double r5860724 = -r5860723;
        double r5860725 = r5860723 * r5860723;
        double r5860726 = 3.0;
        double r5860727 = a;
        double r5860728 = r5860726 * r5860727;
        double r5860729 = c;
        double r5860730 = r5860728 * r5860729;
        double r5860731 = r5860725 - r5860730;
        double r5860732 = sqrt(r5860731);
        double r5860733 = r5860724 + r5860732;
        double r5860734 = r5860733 / r5860728;
        return r5860734;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r5860735 = -0.5;
        double r5860736 = c;
        double r5860737 = b;
        double r5860738 = r5860736 / r5860737;
        double r5860739 = r5860735 * r5860738;
        return r5860739;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

2. Simplified 52.6

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]

3. Taylor expanded around inf 6.1

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

4. Final simplification 6.1

   \[\leadsto -0.5 \cdot \frac{c}{b}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))